Question

Short answer

The maximum chandelier weight that can be supported is \(W = 1173.3\;{\rm{N}}\).

Step-by-step solution

Given data

The angle is\(\theta = 45^\circ \).

The force sustained by the cord is \(F = 1660\;{\rm{N}}\).

Understanding Newton’s second law

Newton’s second law describes the variation that a force generates in the movement of a body. In this question, Newton’s second law for the junction will be used in order to determine the weights in the x and y directions.

Free body diagram and calculation of forces in the x-direction

The following is the free body diagram.

The relation of the forces in the x-direction can be written as:

\(\begin{array}{c}\sum {F_{\rm{x}}} = 0\\\left[ {{F_{\rm{B}}} - {F_{\rm{A}}}\cos \theta } \right] = 0\\{F_{\rm{B}}} = {F_{\rm{A}}}\cos \theta \end{array}\)

Here, \({F_{\rm{A}}}\)and \({F_{\rm{B}}}\) are the forces acting on the chandelier.

In the above relation, it can be seen that force \({F_{\rm{A}}}\) is greater than force \({F_{\rm{B}}}\). Hence, the maximum force will be considered as \({F_{\rm{A}}} = F = 1660\;{\rm{N}}\).

Calculation of forces in the y-direction

The relation of the forces in the y-direction can be written as:

\(\begin{array}{c}\sum {F_{\rm{y}}} = 0\\\left[ {W - {F_{\rm{A}}}\sin \theta } \right] = 0\\W = {F_{\rm{A}}}\sin \theta \end{array}\)

On plugging the values in the above relation, you get:

\(\begin{array}{l}W = \left( {1660\;{\rm{N}}} \right)\sin 45^\circ \\W = 1173.3\;{\rm{N}}\end{array}\)

Thus, \(W = 1173.3\;{\rm{N}}\) is the required weight.